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1. Gradient-Driven Physics Informed Neural Networks for Conduction Heat Transfer and Incompressible Laminar Flow

   https://doi.org/10.1115/1.4070545  

   Lu, Tingying; Shahadat, M_R B; Liu, Qilin; He, Runlin; Jiang, Xiaoyu; Li, Zheng (April 2026, Journal of Computational and Nonlinear Dynamics)

   Abstract Physics-Informed Neural Networks (PINNs) have opened new possibilities for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. However, despite their flexibility, traditional PINNs often struggle to capture sharp gradients and intricate solution features, which limits their effectiveness in many practical problems. In this work, we have introduced Gradient-Driven Physics-Informed Neural Networks (GDPINNs) that improve the ability of traditional PINNs to resolve sharp gradients. By incorporating gradient information directly into the loss function, GDPINNs better target regions where traditional PINNs typically fail. We validated the method on steady-state and transient heat conduction problems, including a central heating source and a sinusoidal boundary condition, and found strong agreement with reference solutions. To further understand the framework's capability, we applied it to a high-gradient steady-state and transient heat conduction problem, where GDPINNs show clear advantages over traditional PINNs and align closely with reference results. We also extended GDPINNs to incompressible laminar flow in a lid-driven cavity, demonstrating its broader applicability. In these cases, GDPINNs consistently provide higher accuracy and better capture critical solution features, highlighting their potential to improve PINNs-based approaches for complex physical problems with sharp gradients. 

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2. Minimizing nature's cost: Exploring data-free physics-informed neural network solvers for fluid mechanics applications

   https://doi.org/10.1063/5.0250022  

   Elmaradny, Abdelrahman; Atallah, Ahmed; Taha, Haithem (February 2025, Physics of Fluids)

   In this paper, we present a novel approach for fluid dynamic simulations by leveraging the capabilities of Physics-Informed Neural Networks (PINNs) guided by the newly unveiled Principle of Minimum Pressure Gradient (PMPG). In a PINN formulation, the physics problem is converted into a minimization problem (typically least squares). The PMPG asserts that for incompressible flows, the total magnitude of the pressure gradient over the domain must be minimum at every time instant, turning fluid mechanics into minimization problems, making it an excellent choice for PINNs formulation. Following the PMPG, the proposed PINN formulation seeks to construct a neural network for the flow field that minimizes Nature's cost function for incompressible flows in contrast to traditional PINNs that minimize the residuals of the Navier–Stokes equations. This technique eliminates the need to train a separate pressure model, thereby reducing training time and computational costs. We demonstrate the effectiveness of this approach through a case study of inviscid flow around a cylinder. The proposed approach outperforms the traditional PINNs approach in terms of training time, convergence rate, and compliance with physical metrics. While demonstrated on a simple geometry, the methodology is extensible to more complex flow fields (e.g., three-dimensional, unsteady, and viscous flows) within the incompressible realm, which is the region of applicability of the PMPG. 

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3. Integrating Physics-Informed Neural Networks and Power Flow for Scalable Online Transient Analysis

   https://doi.org/10.1109/SmartGridComm65349.2025.11204635  

   Gu, Zhangrong; Zhao, Yue; Yue, Meng; Zhao, Tianqiao; Li, Jiaming (September 2025, IEEE)

   Transient dynamic analysis is crucial for power system stability, particularly with increasing renewable integration, yet traditional numerical integration methods are computation- ally expensive for large-scale online applications. We propose a novel framework integrating Physics-Informed Neural Networks (PINNs) with an AC Power Flow (AC-PF) solver to efficiently simulate dynamic trajectories in power systems in a scalable fashion. A separate PINN is trained for each generator to accurately capture the differential equations of the generator dynamic model. The trained PINNs and an AC-PF solver iteratively update the dynamic and algebraic variables to simulate dynamic trajectories for the entire system. The PINN training process consists of an unsupervised stage to enforce physical laws, followed by a supervised stage leveraging simulation data for enhanced accuracy. As the predictor training is performed separately for individual generators, the method’s computational complexity scales linearly with the number of generators in both training and testing. Experiments on a 3-generator 9-bus system demonstrate the very high accuracy of the developed method in simulating full system dynamic trajectories. 

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4. Accelerating Power System Dynamic Simulations with Physics-Informed Neural Networks

   https://doi.org/10.1109/NAPS66256.2025.11272379  

   Sajjadi, Mahsa; Sun, Kai (October 2025, IEEE)

   Physics-informed neural network (PINN) has emerged as a powerful tool to solve the challenging power system problems by incorporating the underlying physical models. This paper uses PINN for solving the differential equations governing a power system model for the Northeast Power Coordinating Council (NPCC) system as a case study. This approach adaptively transitions between the original nonlinear model, a trained PINN, and a hybrid model, which all partition the system into the study area keeping key generators in detail and the simplified external area. It utilizes a fully nonlinear model during the fault-on period, applies a hybrid model for significant rotor angle deviations, and relies on the PINN for smaller state variations. The simulation results are compared with the test cases using the original nonlinear model and using a partitioned hybrid model. PINNs provide faster and more accurate simulation results compared to conventional approaches, enabling efficient computation of dynamic states with reduced reliance on extensive training data. 

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5. Bacteria hinder large-scale transport and enhance small-scale mixing in time-periodic flows

   https://doi.org/10.1073/pnas.2108548118  

   Ran, Ranjiangshang; Brosseau, Quentin; Blackwell, Brendan C.; Qin, Boyang; Winter, Rebecca L.; Arratia, Paulo E. (October 2021, Proceedings of the National Academy of Sciences)

   Understanding mixing and transport of passive scalars in active fluids is important to many natural (e.g., algal blooms) and industrial (e.g., biofuel, vaccine production) processes. Here, we study the mixing of a passive scalar (dye) in dilute suspensions of swimming Escherichia coli in experiments using a two-dimensional (2D) time-periodic flow and in a simple simulation. Results show that the presence of bacteria hinders large-scale transport and reduces overall mixing rate. Stretching fields, calculated from experimentally measured velocity fields, show that bacterial activity attenuates fluid stretching and lowers flow chaoticity. Simulations suggest that this attenuation may be attributed to a transient accumulation of bacteria along regions of high stretching. Spatial power spectra and correlation functions of dye-concentration fields show that the transport of scalar variance across scales is also hindered by bacterial activity, resulting in an increase in average size and lifetime of structures. On the other hand, at small scales, activity seems to enhance local mixing. One piece of evidence is that the probability distribution of the spatial concentration gradients is nearly symmetric with a vanishing skewness. Overall, our results show that the coupling between activity and flow can lead to nontrivial effects on mixing and transport. 

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